Free Diffusion Coefficient Calculator

Diffusion and the Diffusion Coefficient

Diffusion is the process by which particles spread from regions of higher concentration to regions of lower concentration through random thermal motion. The diffusion coefficient ($D$) quantifies the ease with which this transport occurs. The Diffusion Coefficient Calculator (a free online tool) allows you to compute $D$ for particles of various shapes using the Einstein-Smoluchowski relation. By entering parameters such as temperature, solvent viscosity, and particle dimensions, you can obtain an accurate diffusion coefficient quickly.

The Einstein‑Smoluchowski Relation

The derivation of the diffusion coefficient starts from Fick’s first law, which describes the particle flux $J$ as a combination of diffusion and external forces:

$$J = -D \frac{\partial c}{\partial x} + \frac{c f}{\xi}$$

where $c$ is the concentration, $f$ an external force, and $\xi$ the friction coefficient. When the net flow is zero (equilibrium), the two contributions balance, leading to:

$$D = \frac{k_B T}{\xi}$$

Here $k_B$ is Boltzmann’s constant ($1.380649 \times 10^{-23}\ \text{J/K}$), $T$ is the absolute temperature (in kelvin), and $\xi$ is the friction coefficient (with units such as s/kg in SI). The formula shows that diffusion is faster at higher temperatures and slower when the friction coefficient is large.

Friction Coefficient for Common Particle Shapes

The friction coefficient depends on the particle’s shape and size, as well as the solvent’s dynamic viscosity $\eta$. Below is a summary of $\xi$ for several geometries:

Shape	Friction coefficient $\xi$
Sphere (low Re)	$6\pi \eta a$
Disk (face on)	$16 \eta a$
Disk (edge on)	$\dfrac{32}{3} \eta a$
Disk (random motion)	$12 \eta a$
Ellipsoid (lengthwise)	$\dfrac{4\pi \eta a}{\ln(2a/b) + 1/2}$
Ellipsoid (sideways)	$\dfrac{8\pi \eta a}{\ln(2a/b) + 1/2}$
Ellipsoid (random motion)	$\dfrac{6\pi \eta a}{\ln(2a/b)}$

In the table, $a$ and $b$ represent the semi-axes of the particle, and $\eta$ is the solvent’s dynamic viscosity. For a spherical particle, substituting the sphere’s friction coefficient into the diffusion formula gives the well‑known Einstein–Stokes relation.

Example Calculation

Consider the Sulfolobus ellipsoid virus 1, an ellipsoidal virus with semi-axes $a = 115\ \text{nm}$ and $b = 78\ \text{nm}$. Assume it undergoes random tumbling in water at $90\,^{\circ}\text{C}$ (which is $363.15\ \text{K}$). The dynamic viscosity of water at this temperature is approximately $0.00089\ \text{Pa·s}$.

For a randomly moving ellipsoid, the friction coefficient is:

$$\xi = \frac{6\pi \eta a}{\ln(2a/b)} = \frac{6\pi (0.00089\ \text{Pa·s})(115 \times 10^{-9}\ \text{m})}{\ln\bigl(2 \times 115 / 78\bigr)} \approx 3.0 \times 10^{-10}\ \text{s/kg}$$

The diffusion coefficient then follows:

$$D = \frac{k_B T}{\xi} = \frac{(1.380649 \times 10^{-23}\ \text{J/K})(363.15\ \text{K})}{3.0 \times 10^{-10}\ \text{s/kg}} \approx 1.7 \times 10^{-11}\ \text{m}^2/\text{s}$$

This result shows the order of magnitude expected for nanoscale particles in water.

Using the Diffusion Coefficient Calculator

The free Diffusion Coefficient Calculator Online implements all the friction‑coefficient formulas shown in the table. To use it:

1. Select the shape of the particle (e.g., sphere, disk, ellipsoid).
2. Enter the required dimensions (such as radius or semi‑axes).
3. Provide the solvent’s dynamic viscosity and the absolute temperature.
4. The calculator returns the diffusion coefficient instantly.

The tool can also be used in reverse: if you have a measured diffusion coefficient, you can infer the friction coefficient, which is often easier to obtain for molecular‑sized particles.